The Mandelbrot set is built by iterations of function (map)

*z _{m+1} = f( z_{m} ) =
z_{m}^{2} + c* or

for complex

For real c and z are real too
and we can trace iterations on 2D _{o} , z_{m}(x,y) plane. To plot the first
iteration we draw vertical red line from x toward blue
curve _{o} = 0y = f(x) = x, where ^{2} + cy.
_{1} =
f(x_{o}) = c
| |

To get the second iteration we draw red horizontal line to the green
y = x line, where x. Then draw again vertical line to the blue curve
to get _{1} = y_{1} =
f(x_{o})y and so on. Dependence of
_{2} = f(x_{1})x on _{m}m is plotted in the right part of this applet.
Points f at some value _{c}: x_{o} -> x_{1} -> x_{2} ->
...c and x form an _{o}orbit of
x.
_{o} | |

After 25 steps iterations go near to an attracting
fixed point x of the map
_{*} = f(x_{*})f. f doesn't move the point. Fixed points correspond to
intersections of y = x and y = f(x) (green and blue) curves.
There are always two fixed points for a quadratic map f because of
two roots of quadratic equation
f(x
_{*}) - x_{*} =
x_{*}^{2} + c - x_{*} = 0,
x _{1,2} = 1/2 -+ (1/4 - c)^{1/2}The second fixed point (the right intersection) is repelling. | |

The roots may be complex for some c values.
You can see here that for c > 1/4
attracting fixed point becomes repelling (and even complex
therefore we can not see intersections on real plane). Iterations go
to infinity. It is proven that an orbit go to infinity if |z.
Such qualitative change in iteration dynamics is called _{n}|
> 2bifurcation.
Points on complex plane
c that starting
from z don't go to infinity under iterations
form the _{o} = 0Mandelbrot set. | |

For c = -0.75 attracting fixed point becomes repelling and
iterations converge to attracting period-2 orbit x
(see Birth of attracting period 2 orbit later).
_{1}
-> x_{2} -> x_{1} ... |

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